Identi cation and Estimation of Production Function with...

Identification and Estimation of Production Function

with Unobserved Heterogeneity

Hiroyuki Kasahara∗

Vancouver School of EconomicsUniversity of British Columbia

[email protected]

Paul SchrimpfVancouver School of EconomicsUniversity of British Columbia

[email protected]

Michio SuzukiFaculty of EconomicsUniversity of Tokyo

[email protected]

March 21, 2017

Abstract

This paper examines non-parametric identifiability of production function when production

functions are heterogenous across firms beyond Hicks-neutral technology terms. Using a finite

mixture specification to capture unobserved heterogeneity in production technology, we shows

that production function for each unobserved type is non-parametrically identified under regu-

larity conditions. We estimate a random coefficients production function using the panel data of

Japanese publicly-traded manufacturing firms and compare it with the estimate of production

function with fixed coefficients estimated by the method of Gandhi, Navarro, and Rivers (2013).

Our estimates for random coefficients production function suggest that there exists substantial

heterogeneity in production function coefficients beyond Hicks neutral term across firms within

narrowly defined industry.

1 Introduction

Estimation of production function is one of the most important topics in empirical economics.

Understanding how the input is related to the output is a fundamental issue in empirical industrial

organization (see, for example, Ackerberg, Benkard, Berry, and Pakes, 2007). In empirical trade

and macroeconomics, researchers are often interested in estimating production function to obtain

a measure of total factor productivity to examine the effect of trade policy on productivity and

∗Address for correspondence: Hiroyuki Kasahara, Vancouver School of Economics, University of British Columbia,6000 Iona Dr., Vancouver, BC, V6T 1L4 Canada.

1

to analyze the role of resource allocation on aggregate productivity (e.g., Pavcnik, 2002; Kasahara

and Rodrigue, 2008; Hsieh and Klenow, 2009).

As first discussed by Marschak and Andrews (1944), the ordinary least square estimates of

production function suffers from simultaneity bias because inputs are correlated with error term

when a firm makes an input decision based on their productivity level (Griliches and Mairesse, 1998).

Under the assumption that error terms could be decomposed into permanent and idiosyncratic

components, fixed effects estimator may be used but such an assumption could be violated in

practice, and, furthermore, the coefficient of inputs that are persistent over time could be severely

biased downward due to measurement errors (Griliches and Hausman, 1986). More recent literature

attempts to address the simultaneity issue by employing dynamic panel approach (Arellano and

Bond, 1991; Blundell and Bond, 1998; Blundell and Bond, 2000) or developing proxy variable

approach (Olley and Pakes, 1996 (OP, hereafter); Levinsohn and Petrin, 2003 (LP, hereafter);

Ackerberg, Caves, and Frazer, 2006, (ACF, hereafter); Wooldridge, 2009), which are now widely

used in empirical applications.

Despite their popularity, however, potential identification issues of proxy variable approach have

been pointed out in the literature. Bond and Sderbom (2005) and ACF discuss identification issue

due to collinearity under two flexible inputs (i.e., material and labor) in Cobb-Douglas specification.

Gandhi, Navarro, and Rivers (2013, GNR hereafter) argue that, if the firm’s decision follows a

Markovian strategy, then the conditional moment restriction implied by proxy variable approach

may not provide enough restriction for non-parametrically identifying gross production function.

GNR exploit the first order condition with respect to flexible input under profit maximization and

establish the identification of production function without making any functional form assumption.

Based on their identification strategy, GNR proposes an estimation procedure that does not suffer

from simultaneity bias.

This paper extends the identification result of GNR based on the first-order condition to the

case where production functions are heterogenous across firms beyond Hicks-neutral technology

terms. We consider a finite mixture specification in which there are J distinct time-varying pro-

duction technologies and each firm belongs to one of J types. Econometricians do not observe the

type of firms. Without making any functional form assumption on each type of production technol-

ogy, we establish nonparametric identification of J distinct production functions and a population

proportion of each type under the reasonable assumption.

Given that, except for the result of GNR, little formal identification result for production func-

tion estimation in the literature is available, our nonparametric identification result is an important

contribution to the literature. Our identification result on production function with unobserved

heterogeneity is also useful in practice as the random coefficient models for production function

become increasingly popular in empirical analysis (e.g., Mairesse and Griliches, 1990; Van Biese-

broeck, 2003; Doraszelski and Jaumandreu, 2014).

In estimation, we consider a random coefficient specification for production function and propose

two different estimation procedures. The first procedure follows our two-stage identification proof

2

and directly maximizes the log-likelihood function of a finite mixture model of production functions

under parametric assumptions, where the EM algorithm can be used to facilitates the computational

complication of maximizing the log-likelihood function of the finite mixture model. In the second

procedure, we first estimate the partial likelihood function under the normality assumption and

use the posterior distribution of type probabilities to classify each firm observation into one of the

J types, generating J data sets; using each of J data sets, we estimate the rest of the type-specific

parameters. The second procedure is computationally much simpler and requires less auxiliary

parametric assumptions than the first one although the second procedure could lead to a biased

estimator due to misclassification of types when T is small.

We provide empirical evidence that production functions are heterogeneous beyond Hicks-

neutral technology term to motivate the necessity of considering production functions with un-

observed heterogeneity in empirical applications. As analyzed by GNR, if Hicks-neutral technology

term is the only source of permanent unobserved heterogeneity in production function and if in-

termediate input is a flexible input, then we expect that the ratio of intermediate input cost to

output value after controlling for the difference in the input level of capital, labor, and intermedi-

ates should not exhibit any serial correlation. However, using the panel of Japanese manufacturing

firms that belongs to machine industry, we find that the serial correlation of the ratio of inter-

mediate input cost to output value is very high at 0.95 and that, even after controlling for the

difference in the input level of capital, labor, and intermediates, the majority of variation in the

ratio of intermediate input cost to output value can be explained by the firm-specific persistent

component rather than the idiosyncratic component. These findings strongly suggest the presence

of unobserved heterogeneity in production technology beyond Hicks-neutral term within the 3-digit

industry classification.

We estimate a random coefficients production function using the panel data of Japanese publicly-

traded manufacturing firms between 1980 and 2007 and compare the results with those from the

original GNR specification without unobserved heterogeneity. Our estimates suggest that there

exists substantial heterogeneity in production function coefficients beyond Hicks neutral term.

When we estimate production function without incorporating heterogeneity using the estimation

procedure suggested by GNR, we found that the majority of variations in total factor productivity

is coming from idiosyncratic ex-post shocks rather than serially correlated shocks. In contrast,

when we estimate production function with random coefficients, the majority of variations in total

factor productivity is explained by the variation in serially correlated shocks. Furthermore, the

estimated serial correlation in ex-post shocks of random coefficients model is substantially lower

than that of homogenous model. We also find that the correlation between estimated productivity

and investment is different across different types of firms, where the correlation is stronger among

a type of firms with capital intensive production technology than other types of firms.

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2 The Model

Assume that we have panel data of firms i = 1, ..., N over periods t = 1, ..., T for output, the

number of workers, capital, intermediate inputs, and the average wage per worker, denoted by

(Yit,Kit, Lit,Mit,Wit) ∈ Y ×K×L×M×W, respectively. For brevity, let Xit := (Kit, Lit,Mit)′ ∈

X := K×L×M so that (Yit,Kit, Lit,Mit,Wit) = (Yit, Xit,Wit). Each firm’s observation {Yit, Xit,Wit}Tt=1

is randomly sampled from a population distribution P ({Yit, Xit,Wit}Tt=1).

We consider a possibility that firms are different in production technology beyond Hick’s neu-

tral productivity shock. Specifically, we use a finite mixture specification to capture permanent

unobserved heterogeneity in firm’s production technology. Define the latent random variable

Di ∈ {1, 2, ..., J} that represents the type of firm i so that Di = j when firm i has the j-th

type of technology. In the following, the superscript j indicates that functions are specific to type

j while the subscript t indicates that functions are specific to period t. In particular, for a random

variable Zit, we denote the probability distribution and the expectation conditional on Di = j as

P j(Zit) := P (Zit|Di = j) and Ej [Zit] := E[Zit|Di = j]. We assume that both Mit and Lit are

flexibly chosen after observing serially correlated productivity shock ωit and a wage shock vit. On

the other hand, Kit is predetermined at the end of last period. Denote the information available

to a firm for making decisions on Mit and Lit by Iit.

Assumption 1. (a) Each firm belongs to one of J types, where the probability of belonging to type

j is given by πj = P(Di = j), and J is known. (b) For the j-th type of production technology at

time t, the output is related to inputs as

Yit = eωit+εitF jt (Kit, eψjtLit,Mit) = eωit+εitF jt (Kit, Lit,Mit) (1)

where F jt (Kit, Lit,Mit) := F jt (Kit, eψjtLit,Mit), F

jt (·) is a twice continuously differentiable, strictly

increasing, and strictly concave function. Hit := eψjtLit is the labour input in effective unit of labour,

where eψjt ∈ Iit represents the quality of workers for the j-th types of firms relative to other types of

firms with∑J

j=1 πjeψ

jt = 1. (c) The average wage of workers is given by Wit = evit+ζitPH,tHit/Lit =

eψjt+vit+ζitPH,t, where evit+ζitPH,t is the wage per effective unit of labour which is given to firm i at

time t, where PH,t is common across firms.

Assumption 2. (a) (ωit, vit) ∈ Iit. For the j-th type, ωit ∈ Iit follows an exogenous first order

stationary Markov process given by

ωit = hj(ωit−1) + ηit (2)

where, conditional on Iit−1, ηit and vit are mean-zero i.i.d. random variables on R with the proba-

bility density functions gjη(·) and gjv(·), respectively. Furthermore, the unconditional expectation of

ωit is zero, i.e., Ej [ωit] = 0. (b) (εit, ζit) 6∈ Iit so that (εit, ζit) is not known when Lit and Mit are

chosen. For the j-th type, conditional on Iit, (εit, ζit) is a mean-zero i.i.d. random variable on R2

4

with the probability density function gjεζ,t(·).

Assumption 3. (a) Kit ∈ Iit but Kit 6∈ Iit−1. (b) the conditional distribution of Kit given It−1 is

type specific and only depends on Kit−1 and ωit−1, i.e., Pt(Kit|It−1, Di = j) = P jt (Kit|Kit−1, ωit−1).

Assumption 4. (a) Mit and Lit are chosen at time t by maximizing the expected profit conditional

on Iit as

(Mit, Lit) = (Mjt (Kit, ωit),Ljt (Kit, ωit, vit))

:= argmax(M,L)∈M×L

PY,tEj [eεit |Iit]eωitF jt (Kit, L,M)− PM,tM − Ej [eζit |Iit]eψ

jt+vitPH,tL.

(b) Mit is a type-specific deterministic function of Kit and ωit that can be written as Mit =

Mjt (Kit, ωit), where Mj

t is strictly increasing in ωit for any Kit. (c) Lit is a type-specific deter-

ministic function of Kit, ωit, and vit that can be written as Lit = Ljt (Kit, ωit, vit), where Ljt is

strictly decreasing in vit for any (Kit, ωit).

Assumption 5. (a) A firm is a price taker. (b) The intermediate input price PM,t and the output

price PY,t at time t are common across firms. (c) (PM,t, PY,t, PH,t) ∈ Iit and (PM,t, PY,t) is known

to an econometrician.

In Assumption 1, as indicated by the subscript t in F jt (·), type-specific production function

could be different across periods because of type-specific aggregate shocks or type-specific biased

technological changes. The quality of workers also differ across types and periods as captured by the

parameter ψjt , which leads to the systematic difference in the average wage of workers across types.

The restriction∑J

j=1 πjeψ

jt = 1 is necessary for identification of PH,t. The firms are subject to

productivity shocks and wage shocks represented by ωit+ εit and vit+ζit, respectively. Assumption

2 assume that (ωit, vit) is known when Lit and Mit are chosen while (εit, ζit) is not known when

Lit and Mit are chosen. The presence of i.i.d. wage shock vit provides an additional source of

variation for Lit beyond ωit and Kit; consequently, Lit and Mit are not collinear, preventing the

identification problem discussed by Bond and Sderbom (2005) and ACF. The assumption that

Ej [ωit] = 0 is necessary for identification because the production function, F jt (·), differs across

times.

Assumption 3(a) assumes that Kit is determined at time t − 1 so that (ηit, ωit, vit) is not

known when Kit is chosen. Assumption 3(b) can be justified by explicitly considering the dy-

namic model of investment decisions. Assumption 4(b) is a consequence of the strict concavity

assumption in Assumption 1, implying that there exists one-to-one relationship between (Mit, Lit)

and (ωit, vit) conditional on the value of Kit. We may write ωit = Mjt

−1(Kit,Mit) and vit =

Ljt−1

(Kit,Mjt

−1(Kit,Mit), Lit), where Mj

t

−1and Ljt

−1are inverse functions of Mj

t and Ljt with

respect to ωit and vit, respectively.

Under Assumption 5(b), the intermediate input price PM,t cannot be used for instrumentingMit;

when intermediate prices are exogenous and heterogenous across firms, production function could

5

be identified using the intermediate input prices as instruments (see Doraszelski and Jaumandreu,

2014). In Assumption 5(c), we may alternatively assume that a firm is subject to idiosyncratic

price shock ξit such that, for example, PY,it = exp(ξit)PY,t with ξit 6∈ Iit, then ξit plays the similar

role to εit. We may assume that (PM,t, PY,t) is not known to econometrician by treating PM,t/PY,t

as parameters to be estimated; in such a case, we may identify the production function up to scale.

Under Assumptions 1-5, the information set Iit is given by Iit = {ωit, vit,Kit, PH,t, PM,t, PY,t, Vit−1, Vit−2, ...},where Vit = {ζit, εit, ωit, vit,Kit, PH,t, PM,t, PY,t}.

Let gε,t(ε) :=∫gεζ,t(ε, ζ)dζ and gζ,t(ζ) :=

∫gεζ,t(ε, ζ)dε. Under Assumptions 1, 2, 3(a), 4(a),

and 5, the first order conditions with respect to Mit and Lit give

PY,tFjM,t(Xit)E

jt (e

ε)eωit = PM,t, PY,tFjL,t(Xit)E

jt (e

ε)eωit = PH,tEjt (e

ζ)eψjt+vit , (3)

where F jM,t(X) :=∂F jt (X)∂M , F jL,t(X) :=

∂F jt (X)∂L , Ejt [e

ε] :=∫eεgjε,t(ε)dε, and Ejt [e

ζ ] :=∫eζgjζ,t(ζ)dζ.

Equations (1) and (3) give a system of equations

lnYit = lnF jt (Xit) + ωit + εit,

lnSmit = ln(GjM,t(Xit)E

jt [e

ε])− εit,

lnSìt = ln(GjL,t(Xit)E

jt [e

ε]/Ejt [eζ ])− εit + ζit,

(4)

where

Smit :=PM,tMit

PY,tYit, Sìt :=

WitLitPY,tYit

, GjM,t(Xit) :=F jM,t(Xit)Mit

F jt (Xit), and GjL,t(Xit) :=

F jL,t(Xit)Lit

F jt (Xit).

In place of Assumption 5, we may alternatively consider the case where firms produce differen-

tiated products and face a demand function with constant price elasticity as follows.

Assumption 6 (Constant Demand Elasticity). (a) A firm faces an inverse demand function with

constant elasticity given by PY,it = Y−1/σjYit eε

jd,it, where εd,it /∈ Iit is an i.i.d. ex-post shock that is

not known when Mit is chosen at time t. (b) A firm is a price taker for intermediate and labour

inputs and the intermediate and labour input prices at time t, PM,t and PL,t, are common across

firms. (c) (PL,t, PM,t, PY,t) ∈ Iìt ⊂ Iit. (d) PY,it and Yit are not separately observed in the data.

Under Assumption 6, the “revenue” production function is given by PY,itYit = F jt (Xit)eωit+εit ,

where F jt (Xit) := [F jt (Xit)]

σjY

−1

σjY , ωit :=

σjY −1

σjYωit, ζit :=

σjY −1

σjYζit, and εit := εdit +

σjY −1

σjYεit. Then, in

place of (4), we have

lnPY,itYit = ln F jt (Xit) + ωit + εit,

lnSmit = ln(GjM,t(Xit)E

jt [e

ε])− εit,

lnSìt = ln(GjL,t(Xit)E

jt [e

ε]/Ejt [eζ ])− εit + ζit,

(5)

6

where GjM,t(Xit) :=F jM,t(Xit)Mit

F jt (Xit)and GjL,t(Xit) :=

F jL,t(Xit)Lit

F jt (Xit). When PY,it and Yit are not separately

observed in the data, the observable implication of (5) are the same as that of (4). In particular, we

cannot separately identify the parameter σjY and the production function F jt . Therefore, we focus

on the identification analysis under Assumption 5 although we should be careful in interpreting the

empirical result because the unobserved heterogeneity in revenue production function could partly

reflect in difference in demand elasticity.

3 Nonparametric identification

In this section, we establish the non-parametric identification of production functions with unob-

served heterogeneity using the second and third equations of (4) as an additional restriction. For

notational brevity, we drop the subscript i in this section and denote St = (Smt , S`t ). Note that, by

definition of S`t and Smt , we have Wt =S`tPM,tMt

Smt Ltand Yt =

PM,tMt

Smt PY,tso that the value of Wt and Yt

is known given (St, Xt) under Assumption 5. Therefore, we consider {St, Xt}Tt=1 as our data. Let

Zt := (St, Xt) ∈ S × X .

We first establish the nonparametric identification of model structures when J = 1 as follows.

Proposition 1. Suppose that J = 1 and Assumption 1-5 holds with T ≥ 3. Then, (a) θ1 :=

{gv(·), gεζ,t(·), GM,t(·), GL,t(·), PH,t}Tt=1 is uniquely determined from P({Zt}Tt=1). (b) θ2 := {{Ft(·)}Tt=2, h(·), gη(·)}is uniquely determined from P({Zt}Tt=1) and θ1.

Remark 1. Proposition 1 extends the identification result of GNR to the setting where Lit is

contemporaneously determined rather than predetermined.

When J ≥ 2, the distribution of {Zt}Tt=1 follows an J-term mixture distribution

P({Zt}Tt=1) =J∑j=1

πjPj1(Z1)T∏t=2

Pjt (Zt|{Zt−s}t−1s=1). (6)

Proposition 2. Suppose that Assumptions 1-5 hold. Then, the distribution of {Zt}Tt=1 defined in

(6) can be written as

P({Zt}Tt=1) =J∑j=1

πj

(Pj1(S1|X1)

T∏t=2

Pjt (St|Xt)

)×

(Pj1(X1)

T∏t=2

Pjt (Xt|Xt−1)

). (7)

Therefore, {Zt}Tt=1 follows a first order Markov process within subpopulation specified by type.

The result of Proposition 2 allows us to establish the nonparametric identification of {πj , {Pjt (Zt)}Tt=1}Jj=1

by extending the argument in Kasahara and Shimotsu (2009) and Hu and Shum (2012).

Assumption 7. Let Wt be the support of Wt. For every (z2, z3) ∈ Z2 ×Z3, there exists (z2, z3) ∈Z2 × Z3, (a1, ..., aJ) ∈ ZJ1 and (b1, ..., bJ−1) ∈ ZJ−1

4 such that (a) Lz3, Lz3, Lz2, and Lz2 defined

7

in (33) are nonsingular, (b) P j(Z3 = z3|Z2 = z2) 6= 0 and P j(Z3 = z3|Z2 = z2) 6= 0 hold for

j = 1, ..., J , and (c) all the diagonal elements of Dz2,z2,z3,z3 defined in (34) take distinct values.

Proposition 3. Suppose that Assumptions 1-5, and 7 hold and T ≥ 4. Then,

{πj ,Pj1(Z1), {Pjt (Zt|Zt−1)}Tt=2}Jj=1 is uniquely determined from P({Zt}Tt=1).

Remark 2. Under the additional assumption of the stationarity, i.e., Pjt (Zt|Zt−1) = Pj(Zt|Zt−1)

for t = 2, ..., T , Kasahara and Shimotsu (2009) establishes the nonparametric identification of the

model (7) when T = 6 while Hu and Shum (2013) shows that T = 4 suffices for identification.

Remark 3. Considering serially correlated continuous unobserved variables {X∗t }, Hu and Shum

(2013) analyze the nonparametric identification of the model

P({Zt}Tt=1) =

∫P1(Z1, X

∗1 )

T∏t=2

Pt(Zt, X∗t |Zt−1, X

∗t−1)d({X∗t }Tt=1).

Given the panel data {Zt}Tt=1 with T = 5, Theorem 1 and Corollary 1 of Hu and Shum (2013) state

that, under their Assumptions 1-4, P3(Z3, X∗3 ), P4(Z4, X

∗4 |Z3, X

∗3 ), and P5(Z5, X

∗5 |Z4, X

∗4 ) are non-

parametrically identified but the identification of P1(Z1, X∗1 ), P2(Z2, X

∗2 |Z1, X

∗1 ), and P3(Z3, X

∗3 |Z2, X

∗2 )

remains unresolved. Our Proposition 3 shows that, for a model in which unobserved heterogeneity

is discrete and finite, we can nonparametrically identify the type-specific distribution of {Zt}Tt=1

including the first two periods of the data from T = 4 periods of panel data without imposing

stationarity.

Remark 4. Assumption 7 assumes the rank condition of matrices Lz3, Lz3, Lz2, and Lz2 defined

in (33), of which elements are constructed by evaluating Pj4(Z4|Z3) and πjPj2(Z2|Z1)Pj1(Z1) at dif-

ferent points. These conditions are similar to the assumption stated in Proposition 1 of Kasahara

and Shimotsu (2009). Please refer to Remark 2 of Kasahara and Shimotsu (2009) for their inter-

pretations. One needs to find only one pair of values (Z2, Z3) ∈ Z2 ×Z3 and one set of J − 1 and

J points of Z1 and Z4 to construct nonsingular Lz3, Lz3, Lz2, and Lz2 for each (Z2, Z3) ∈ Z2×Z3

and these rank conditions are not stringent when Wt has continuous support. The identification

of P j4 (Z4|Z3 = Z3) and πjP j2 (Z2 = Z2|Z1)P j1 (Z1) at all other points of Z4 and Z1, respectively,

follows without any further requirement on the rank condition.

Once the type-specific distribution of {Zt} is identified, we may apply the argument in the proof

of Proposition 1 to prove the nonparametric identification for each type’s model structure.

Proposition 4. Suppose that Assumptions 1-5, and 7 hold and T ≥ 4. Then, (a) θ1 :=

{πj , gjv(·), {gjεζ,t(·), , GjM,t(·), G

jL,t(·), PH,t, ψ

jt }Tt=1}Jj=1 is uniquely determined from P({Zt}Tt=1). (b)

θ2 := {{{F jt (·)}Tt=1}Jj=2, hj(·), gjη(·)} is uniquely determined from P({Zt}Tt=1) and θ1.

Therefore, type-specific production functions as well as the distribution of unobserved variables

can be non-parametrically identified. In estimation, we focus our attention to the case where

type-specific function is given by Cobb-Douglas production function with random coefficients.

8

Example 1 (Random Coefficients Model). Consider a Cobb-Douglas production function with

time-varying random coefficients:

lnF jt (Xt) = βj0,t + βjk,tkt + βj`,t`t + βjm,tmt, (8)

where the intermediate and labor cost share equations are given by

lnSmt = ln(βjm,t) + lnEjt [eε]− εt,

lnS`t = ln(βj`,t) + ln(Ejt [e

ε]/Ejt [eζ ])− εt + ζt.

Under Assumptions 1-5, and 7, {πj , hj(·), gjη(·), gv(·), {βj`,t, βjm,t, g

jε,t(·), g

jζ,t(·), ψ

jt }4t=1, {β

j0,t, β

jk,t}

4t=2}

for j = 1, ..., J is nonparametrically identified from the panel data {St, Xt}4t=1.

In the appendix, we discuss the conditions under which Assumption 7 holds when the production

function is Cobb-Douglas. The following corollary shows that type-specific distribution of St can

be identified from the joint distribution of {St}Tt=1 for Cobb-Douglas specification.

Corollary 1. Suppose that Assumptions 1-5, and 7 hold and T ≥ 4. Suppose that production

function is Cobb-Douglas given by (8). Then, {πj , {Pjt (St)}Tt=1}Jj=1 is uniquely determined from

P({St}Tt=1).

4 Estimation of production function with random coefficients

We consider a random sample of N independent observations {{Yit, Xit, Sit,Wit}Tt=1}Ni=1 from the

J-component mixture model∑J

j=1 πjPjt ({Yit, Xit, Sit,Wit}Tt=1) =

∑Jj=1 π

jPjt ({Sit, Xit}Tt=1).

We impose the following parametric assumptions for estimation.

Assumption 8. (a) Assumption 1 holds with

Yit = F jt (Kit, Lit,Mit)eωit+εit with F jt (Kit, Lit,Mit) = exp(βj0,t + βjkkit + βj` ìt + βjmmit). (9)

(b) Assumption 2 holds with

(εit

ζit

)∣∣∣∣∣Di = jd∼ N

((0

0

),

((σjε )2 ρjεζσ

jεσ

jζ

ρjεζσjεσ

jζ (σjζ)

2

)).

In (9), because logHit = ψjt + ìt, the parameter βj0,t contains β`ψjt and captures the differ-

ence in the quality of workers across types. The normality assumption in Assumption 8(b) can

be potentially relaxed, for example, using the maximum smoothed likelihood estimator of finite

mixture models of Levine et al. (2011) in which the type-specific distribution of εit and ζit is non-

parametrically specified. Kasahara and Shimotsu (2015) develop a likelihood-based procedure for

testing the number of components in normal mixture regression models.

Denote the log values of (Yit, Lit,Kit,Mit, Smit , S

ìt,Wit) by (yit, ìt, kit,mit, s

mit , s

ìt, wit) and let

sit := (sìt, smit ) and xit := (ìt, kit,mit). Under Assumptions 3-5, 8, the first order conditions for the

9

expected profit maximization imply that

smit = lnβjm + 0.5(σjε )2 − εit and s`it = lnβj` + 0.5

{(σjε )

2 − (σjζ)2}− εit + ζit. (10)

We propose two different estimation procedures. The first procedure directly maximizes the log-

likelihood function of a finite mixture model of production functions under additional parametric

assumptions on the law of motion for kit and the initial distribution of (kit, ωit), where the likelihood

function is a parametric version of (7). Because the maximum likelihood estimator utilizes the

distributional information, it is consistent even when T is small as long as T ≥ 4. Our estimation

procedure follows the two-stage identification proof of Proposition 3. The EM algorithm can be

used to facilitates the computational complication of maximizing the log-likelihood function of the

finite mixture model.

In the second procedure, we first estimate the partial likelihood function of the input share

equations (10) under the normality assumption and use the posterior distribution of type probabil-

ities to classify each firm observation into one of the J types under the assumption that T → ∞.

This generates J data sets, where a firm’s production technology becomes increasingly homoge-

nous within each of the J data sets as T → ∞. In the second stage, we estimate the rest of the

type-specific parameters by using each of J data sets.1

The first procedure can consistently estimate the parameter even when T is small as long

as T ≥ 4 and N → ∞ but it is computationally more complicated and requires more auxiliary

parametric assumptions than the second one. We introduce the second procedure because it is

computationally much simpler than the first one although, when T is small, the second procedure

leads to a biased estimator due to misclassification of types.

4.1 Maximum likelihood estimator

We make the parametric distributional assumptions and develop parametric maximum likelihood

estimator.

Assumption 9. (a) T is fixed at T ≥ 4 and N →∞. (b) Assumption 2 holds with hj(ωit) = ρjωωit

so that

ωit = ρjωωit−1 + ηit, (11)

gjη(η) = φ(η/σjη)/σjη, and gjv(v) = φ(v/σjv)/σ

jv. (c) Assumption 3 holds with the additional assump-

tion that, conditional on being type j, kit given (kit−1, ωit−1) is normally distributed with mean

ρjk0 + ρjkkkit−1 + ρjkωωit−1 and variance (σjk)2 while the distribution of (ki1, ωi1) follows a bivariate

normal distribution with mean µj1 and variance Σj1.

1Note that the identification of production function immediately follows from T → ∞ without appealing toProposition 3 because, in principle, each firm’s production function can be identified from the time-series data ofeach firm.

10

Collect the model parameters into θ1, and θ2 as follows. Let

θ1 = (π′, θ11, ..., θ

J1 )′, θ2 = ((θ1

2)′, ..., (θJ2 )′)′, and θj = ((θj1)′, (θj2)′)′ where

θj1 = (βjm, βj` , (σ

jε )

2, (σjζ)2)′ and θj2 = (βj2, ..., β

jT , β

jk, (µ

j1)′, vech(Σj

1)′, ρjk0, ρjkk, ρ

jkω, σ

2k, ρ

jω, σ

jη)′.

In view of Proposition 2 and equation (10), we may write the probability density function of

{sit, xit}Tt=1 for type j as

fjt ({sit, xit}Tt=1) =

T∏t=1

fjt (sit; θj1)︸︷︷︸

=L1i(θj1)

× fj1(xi1; θj)

T∏t=2

fjt (xit|xit−1; θj)︸︷︷︸=L2i(θ

j1,θ

j2)

,(12)

where the exact expression for L1i(θj1) and L2i(θ

j2, θ

j1) is derived below.

Given the decomposition (12), we estimate the model by two-stage maximum likelihood estima-

tion procedure. In the first stage, we estimate π and θ1 by maximizing∑N

i=1 log(∑J

j=1 πjL1i(θ

j1))

over π and θ1. In the second stage, we estimate θ2 given the first stage estimate π and θ1 by

maximizing∑N

i=1 log(∑J

j=1 πjL1i(θ

j1)L2i(θ

j1, θ

j2)) over θ2.

From equation (10), we can compute εit and ζit as

ε∗(sit; θj1) := −smit + lnβjm + 0.5(σjε )

2 (13)

ζ∗(sit; θj1) := sìt − smit − ln(βj`/β

jm) + 0.5(σjζ)

2. (14)

In the first stage, we estimate θ1 by the maximum likelihood estimator given by

θ1 = argmaxθ1

N∑i=1

ln

J∑j=1

πjL1i(θj1)

with

L1i(θj1) :=

T∏t=1

1√1− (ρjεζ)

2σjεσjζ

φ

(ε∗(sit; θ

j1)

σjε

)φ

ζ∗(sit; θj1)− ρjεζ(σjζ/σ

jε )ε∗(sit; θ

j1)√

1− (ρjεζ)2σjζ

.

In the second stage, from (9), εt = E[smt |xt]− sm, and yt + smt = mt + ln(PM,t/PY,t), we have

ωit = ω∗t (mit, ìt −mit, kit; θj) := (1− βjm − β

j` )mit − βjt − β

j` (ìt −mit)− αjkkit, (15)

where βjt = β0,t − ln(PM,t/PY,t). Furthermore, because vit = wit − (ψjt + lnPH,t + ζ∗(sit; θj1)) and

sìt − smit = (wit + ìt)− (lnPM,t +mit), we have

vit = v∗t (ìt −mit; θj) := −(ìt −mit + ψjt − ln(βj`/β

jm) + 0.5(σjζ)

2), (16)

where ψjt := ψjt + ln(PH,t/PM,t).

11

By the change of variables in equation (15), we can relate the density function of mit conditional

on ìt − mit and kit to the density function of ωit, denoted by gω,t, as fjt (mit|ìt − mit, kit) =

(1−βjm−βj` )gjω,t(ω

∗t (mit, ìt−mit, kit; θ

j)). Similarly, we can relate the density function of ìt−mit

to that of vit. Then, from (15)-(16) and Assumptions 2-3, we have

fj1(mi1|ì1 −mi1, ki1; θj) = (1− βjm − βj` )g

jω|k,1(ω∗i1(θj)|ki1), (17)

fjt (mit|ìt −mit, kit, xit−1; θj) = (1− βjm − βj` )g

jη(η∗it(θ

j)) for t ≥ 2, (18)

fjt (ìt −mit|kit, xit−1; θj) = fjt (ìt −mit|kit; θj) = gjv(v∗it(θ

j)) for t ≥ 1, (19)

fjt (kit|xit−1; θj) = gjk,t(kit|kit−1, ω∗i,t−1(θj)) for t ≥ 2, (20)

where gjω|k,1(ωi1|ki1) is the density function of ωi1 conditional on ki1, gjk,t(kit|kit−1, ωit−1) is the

density function of kit given (kit−1, ωit−1), ω∗it(θj) := ω∗t (mit, ìt −mit, kit; θ

j), v∗it(θj) := v∗t (ìt −

mit; θj), and

η∗it(θj) := ω∗it(θ

j)− ρjωω∗i,t−1(θj). (21)

Therefore, under Assumption 9, it follows from (12) and (17)-(20) that

L2i(θj) = fj1(mi1|ì1 −mi1, ki1; θj)fj1(ì1 −mi1|ki1; θj)fj1(ki1; θj)

×T∏t=2

fjt (mit|ìt −mit, kit, xit−1; θj)fjt (ìt −mit|kit, xit−1; θj)fjt (kit|xit−1; θj)

=

(T∏t=1

gjv(v∗it(θ

j))

)×

((1− βjm − β

j` )T gjωk,1(ω∗i1(θj), ki1)

T∏t=2

gjη(η∗it(θ

j))gjk,t(kit|kit−1, ω∗i,t−1(θj))

),

where

gjv(v∗it(θ

j)) =1

σjvφ

(v∗it(θ

j)

σjv

), gjη(η

∗it(θ

j)) =1

σjηφ

(η∗it(θ

j)

σjη

),

gjωk,1(ω∗i1(θj), ki1) = (2π)−3/2|Σj1|−1/2 exp

(−1

2

((ki1

ω∗i1(θj)

)− µj1

)′(Σj

1)−1

((ki1

ω∗i1(θj)

)− µj1

)),

gjk,t(kit|kit−1, ω∗i,t−1(θj)) =

1

σjkφ

(kit − (ρjk0 + ρjkkkit−1 + ρjkωωit−1)

σjk

).

Given the first stage estimate θ1, the parameter π and θ2 can be estimated by maximizing the

log-likelihood function as

(π, θ2) = argmaxπ,θ2

N∑i=1

log

J∑j=1

πjL1i(θj1)L2i(θ

j1, θ

j2)

.

In practice, we use EM algorithm to estimate θ1, θ2, and π as discussed in the Appendix.

12

4.2 Estimation by classifying each observation into one of the J types

Given the first stage estimate θ1, define the posterior probability of being type j for each firm i by

πji =πjL1i(θ

j1;T )∑J

k=1 πkL1i(θk1 ;T )

for j = 1, ..., J , (22)

where we explicitly write the dependence of the likelihood on the length of panel data T in

L1i(θk1 ;T ). We classify each firm into one of the J types by taking the type that gives the highest

posterior probability as its type. Then, for each i, our estimator of Di is given by

Di = argmaxj=1,..,J

{πji }.

Denote the true value of θj1 by θj∗1 . We assume that T → ∞ but require that T goes to ∞ at

much slower rate than N .

Assumption 10. N,T →∞ and√N

exp(ajT )/√T→ 0 for j = 1, ..., J , where aj = mink 6=j E[lnL1it(θ

j∗1 )−

lnL1it(θk∗1 )|i ∈ Ij ] > 0.

Proposition 5. For each i ∈ Ij, πji − 1 = op(N−1/2) under Assumption 10.

Proposition 5 implies that, when Assumption 10 holds, the possible classification error across

types does not affect our inference.

In the second stage, we compute the estimate of ηjit for t = 2, ..., T for each a candidate value of

θj2 given the first stage estimate θj1 as in (21) using the subsample of firms for which Di = j. Then,

stacking the moment conditions implied by E[η∗it(θj1, θ

j2)|kit, xit−1] = 0 for t = 2, ..., T , we can use

standard GMM procedure to estimate θj2 as

θj2 = argminθ2

1

#{i : Di = j}

∑i∈{i:Di=j}

gi(θ2)

1

#{i : Di = j}

∑i∈{i:Di=j}

gi(θ2)

′ for j = 1, ..., J ,

where #{i : Di = j} is the number of firms with Di = j while gi(θ2) := (η∗i2(θj1, θj2)Zi2(θj1, θ

j2)′, ...,

η∗iT (θj1, θj2)ZiT (θj1, θ

j2)′)′ with Zit(θ

j1, θ

j2) := (1, kit, ω

∗it−1(θj1, θ

j2))′.

5 Empirical applications

5.1 Data

We use Japanese publicly traded manufacturing firms, 1980-2007. The data set compiled by the

Development Bank of Japan (DBJ) contains detailed corporate balance sheet/income statement

data for the firms listed on the Tokyo Stock Exchange.2 The initial value of capital (K) is defined

2Because firm’s financial data do not necessarily refer to a calendar year, we assign year t to an observation if thegiven firm’s closing date is between June of year t and May of year t + 1. If firms change their closing dates, the

13

as fixed asset less land from the firm’s balance sheet and the subsequent values of capital are

constructed by perpetual inventory method. The labor input (L) is the number of employees. The

intermediate input (M) is defined as the sum of energy input, material input, transportation cost,

outsourcing cost, and changes in input inventories. The output (Y ) is defined as the value of total

sales plus the changes in inventories of finished goods. The machine investment rate (Im,itKm,it

) is

defined as the ratio of machine investment to machine capital stock. In this preliminary version,

we focus on a sample from Machine industry. Table 1 presents summary statistics for the variables

we use in our empirical analysis.

Table 1: Summary statistics

Statistic N Mean St. Dev. Min Max

lnYit 5602 17.108 1.368 12.191 21.785lnMit 5602 16.314 1.472 9.003 21.306lnLit 5602 6.647 1.189 2.890 10.978lnKit 5602 15.926 1.415 12.223 21.328

lnPM,tMit

PY,tYit5602 -0.777 0.510 -6.930 1.708

IitKit

5602 0.100 0.151 -0.491 2.849

5.2 Evidence for unobserved heterogeneity

In the data, the material share is heterogenous across firms and persistent over time within firm.

Figure 1 presents the histogram ofPM,tMit

PY,tYitacross all observations that belongs to Machine industry,

which shows a large variation in material shares. In the model, the variation in material shares is

coming from idiosyncratic ex-post shocks εit. We may eliminate most of idiosyncratic components

by considering the firm-level average of material shares over 28 years; however, in Figure 2, the

persistent component of the ratio of intermediate inputs to total sales substantially varies across

firms. As shown in Figures 3 and 4, we also observe a large variation in the persistent component

in the ratio of intermediate cost to the sum of intermediate cost and total wage bills,PM,tMt

PM,tMit+WtLit,

of which variation is not likely to be driven by a variation in markups.

Figure 5 plots each firm’s material share, output, and inputs from 1980 to 2007. The presence

of heterogeneity across firms and the persistence within each firm in materials shares are apparent

in the upper left panel of Figure 5. It also appears that labour and capital inputs are changing over

time more smoothly than material input, suggesting that material input responds to idiosyncratic

shocks more than labour and capital inputs do within a short period of time. As shown in Figure

6, the heterogeneity in material shares across firms do not disappear even when we examine firms

within the subindustries of Machine industry, which roughly corresponds to 4-digit ISIC.

data after the change may refer to less than 12 months. When it occurs, we multiply the data xit by 12/m where mrepresents the number of months to which the data refer.

14

In view of the intermediate share equation in (4), a large cross-sectional variation in the per-

sistent component of the ratio of intermediate inputs to total sales suggests either heterogeneity in

production function or the persistence in inputs over time. To examine further, we regress lnSit

on the third order polynomials of (lit, kit,mit) to get residuals, denoted by eit, and decompose eit

into permanent components and idiosyncratic components as ξi := T−1∑T

t=1 eit and ζit := eit− ξi.Comparing the variance of ξi with that of ζit, we found that Var(ξi)

Var(ξi)+Var(ζit)= 0.612. Therefore, the

majority of variation is coming from the permanent component even after controlling the observed

input (lit, kit,mit), suggesting that production function is heterogeneous beyond Hicks-neutral term.

We also estimated the value added Cobb-Douglas specification

yvait = αvat + αva` `it + αvak kit + ωvait + εvait ,

by the approach developed by Levinsohn and Petrin, where yvait is the logarithm of value added.

When we compute the serial correlation of estimated values of εvait , we found that the correlation

coefficient of 0.85.3 One possible reason for this high correlation of estimated values of εvait is the

presence of unobserved heterogeneity in (αvat , αva` , α

vak ).

5.3 Estimation of production function

Given the relatively long length of our panel data, we apply our proposed estimation method based

on classifying each firm into one of the J types. Table 2 presents the parameter estimates for the

number of components equal to J = 1, 3, and 5. Setting J = 1 gives the homogenous production

function specification considered by GNR.

The estimated coefficients across different types when J = 3 and 5 suggest that there are

substantial differences in the output elasticities with respect to materials, labor, and capital across

firms. For the model with J = 3, the material share is lowest for Type 1 and highest for Type 3

while Type 1 is more labor intensive than Type 2 or 3. For the model with J = 5, the material

share of Type 1 is the highest while the material share of Type 2 is the lowest among three types.

The degree of capital intensity is also different across five types, where Type 5 is the most capital

intensive while Type 2 is the most labor intensive.

Figure 7 shows the distribution of posterior type probabilities, defined in (22), across firms for

the model with J = 3 and 5, respectively. The posterior probabilities for each type are concentrated

on around 0 or 1, which is consistent with the result of Proposition 5 where Assumption 10 could

be roughly applied here given T = 28 in our data set. We assign one of the J types to each firm

based on its posterior type probability that achieves the highest value across J types.

Figure 8 plots each firm’s material share and the log of output from 1980 to 2007, where different

colours represent different types for the model with J = 3. From the left panel of Figure 8, it is

clear that each firm’s type is identified with its average material share. On the other hand, it does

3Using the OP approach with the value-added specification, Fox and Smeets (2007) also report the high serialcorrelation of estimated values of idiosyncratic shocks.

15

Table 2: Estimates of Production Function (9): Machine Industry in Japan, 1980-2008

Estimation by ClassificationGNR Random Coefficients ModelJ = 1 J = 3 J = 5

Type 1 Type 2 Type 3 Type 1 Type 2 Type 3 Type 4 Type 5

βjm 0.340 0.623 0.184 0.438 0.112 0.642 0.387 0.267 0.509

βj` 0.422 0.215 0.845 0.416 0.954 0.200 0.577 0.611 0.357

βjk 0.260 0.162 0.134 0.195 0.097 0.154 0.089 0.230 0.162

βjm + βj` + βjk 1.021 1.000 1.163 1.049 1.163 0.995 1.054 1.109 1.028

βjk/βj` 0.617 0.753 0.159 0.470 0.102 0.772 0.155 0.376 0.455

πj 1.000 0.467 0.200 0.333 0.082 0.373 0.155 0.113 0.277

No. of Obs. 5602No. of firms 240

not appear that there is any systematic difference across types in terms of the distribution of firm

sizes measured by outputs.

Table 3 shows a fraction of firms belonging to each type of the J types within subindustries

of Machine industry for the model with J = 3 and 5. See Figure 6 for the definition of subindus-

tries. The distribution of types is quite different across subindustries. Figure 9 plots each firm’s

material share from 1980 to 2007 where different colours identify firm’s type for the model with

J = 3, suggesting that our framework flexibly captures the unobserved heterogeneity in production

technology within more narrowly defined subindustries.

The first two rows of Table 4 present the standard deviations of ωit + αjt and εit. To compute

the standard deviations of ωit + αjt and εit, we compute ωit, ξi, and εit by assigning one of the

J = 3 types to each firm based on its posterior type probabilities. As the number of components

J increases from 1 to 3, and then to 5, the standard deviations of ωit + αjt and εit within each

type decrease on average across types. This indicates a possibility of substantial upward bias in

the estimated variation in ex-post shocks in homogenous model with J = 1 as a result of ignoring

unobserved heterogeneity.

The third row of Table 4 reports the serial correlation in εit. The serial correlation in εit is very

high at 0.951 when J = 1. Given that the presence of high serial correlation indicates a possibility

of misspecificaiton of the model, a smaller value of serial correlation is more desirable. When the

number of components increases from J = 1 to J = 3, and then to J = 5, the average serial

correlation in εit across types decreases from 0.951 to 0.762, and then to 0.693, indicating that the

very high serial correlation in εit when J = 1 is partly due to ignoring unobserved heterogeneity

in production function coefficients. On the other hand, the level of serial correlation in εit is still

high at 0.693 when J = 5. To examine this issue, we also consider a more flexible model in which

the material share parameter is firm-specific. In this case, the material share equation is given by

firm-fixed effects model: sit = αm,i − εit. The estimated serial correlation in εit for this firm-fixed

16

Table 3: A fraction of firms for each type by subindustry for the model with J = 3 and 5

J = 3 J = 5Type 1 Type 2 Type 3 Type 1 Type 2 Type 3 Type 4 Type 5

2511 0.890 0.000 0.110 0.000 0.890 0.110 0.000 0.0002521 0.734 0.122 0.144 0.045 0.614 0.000 0.077 0.2642522 0.599 0.288 0.114 0.220 0.379 0.000 0.068 0.3332523 0.120 0.776 0.104 0.000 0.120 0.104 0.776 0.0002529 0.000 0.417 0.583 0.000 0.000 0.583 0.417 0.0002531 0.564 0.178 0.258 0.000 0.195 0.000 0.178 0.6282532 0.231 0.391 0.378 0.338 0.102 0.120 0.053 0.3872533 0.481 0.000 0.519 0.000 0.481 0.193 0.000 0.3262534 0.762 0.140 0.098 0.140 0.738 0.000 0.000 0.1222535 0.377 0.449 0.174 0.247 0.237 0.130 0.154 0.2332536 0.506 0.090 0.404 0.008 0.364 0.138 0.082 0.4082537 0.462 0.120 0.418 0.071 0.399 0.165 0.049 0.3162541 0.154 0.215 0.631 0.046 0.133 0.471 0.170 0.181

Notes: Subindustries are 2511: Boiler prime mover, 2521: Metal machine tools, 2522: Metalworking machinery, 2523: Machinery tool, 2531: Textile machinery, 2532: Agricultural machines,2533: Construction and mining equipment, 2534: Chemical machinery, 2535: Office machinery,2536: Special industrial machinery, 2537: General industrial machinery, 2541: General MechanicalComponents.

17

effects model remains quite strong at 0.773, suggesting that the material share parameter could

change over time persistently within the same firm.

Ignoring unobserved heterogeneity may lead to substantial biases in the measurement of produc-

tivity growth. To examine this issue, we take a specification with J = 5 as the true model and com-

pute the bias in the measurement of productivity growth when we use a misspecified model with J =

1. Specifically, let ∆ωit := ∆yit−(αjt+αjm∆mit+α

j`∆ìt+α

jk∆kit+∆εjit) for j = 1, 2, ..., 5 be an esti-

mated productivity growth when J = 5 and let ∆ωit := ∆yit−(αt+αm∆mit+α`∆ìt+αk∆kit+∆εit)

be an estimated productivity growth when J = 1, where {αjt , αjm, α

j` , α

jk}

5j=1 and {αt, αjm, αj` , α

jk}

denote estimated coefficients when J = 5 and J = 1, respectively. Then, we compute the bias as

∆ωit = ∆ωit + (βm − βjm)∆mit + (β` − βj` )∆ìt + (βjk − βjk)∆kit + (∆εit −∆εjit)︸︷︷︸

:=Biasit

.

Table 5 reports the ratio of the average absolute value of bias to the average productivity growth

within each of five subsamples, classified by types. The magnitude of the bias is high on average and

substantially different across different types. In particular, the bias in the measured productivity

growth when J = 1 is larger than 40 percent of the average productivity growth for Type 1

and 2, indicating that ignoring unobserved heterogeneity could result in serious bias in estimated

productivity growth.

As an example of using the estimated productivity growth in empirical analysis, we now ex-

amine whether unobserved heterogeneity captured by type-specific production function parameter

is important for investment decision. Specifically, for each of subsample classified by types, we

estimate the following linear investment model

IitKit

= α0 + αωωit + quadratic of ìt and kit + ζit,

where Iit/Kit denotes the ratio of investment to capital stock.

Table 6 reports the estimated coefficients of ωit across different specifications and different types

for J = 1, 3, and 5. The coefficient of ωit is estimated significantly at 0.016 when J = 1. For the

model with J = 3 and 5, the estimated coefficients of ωit are substantially different across different

types of firms. For J = 3, the coefficient of ωit for Type 2 is insignificant at -0.007 while the

coefficients of ωit are estimated significantly at 0.090 and 0.068 for Type 1 and 3, respectively. As

reported in the second and third rows of Table 6, the material shares of Type 1 and 3 are higher

than that of Type 2 while Type 1 and 3 is more capital intensive than Type 2 for the model with

J = 3. For J = 5, the coefficients of ωit are high at 0.094 and 0.083 for Type 2 and 5, respectively,

both of which have relatively higher material shares and more capital intensive technology than

other three types. Therefore, we find that the correlation between productivity and investment

is stronger among a type of firms with high material shares and high capital intensive production

technology than other types of firms.

18

Tab

le4:

Ran

dom

Coeffi

cien

tsM

od

el(9

):M

ach

ine

Ind

ust

ryin

Jap

an,

1980

-200

8

Est

imat

ion

byC

lass

ifica

tion

J=

1J=

3J=

5Type1

Type2

Type3

Ave†

Type1

Type2

Type3

Type4

Type5

Ave†

Std.Dev.ωit+αj t

0.472

0.195

0.620

0.241

0.295

0.895

0.185

0.307

0.359

0.190

0.283

Std.Dev.ε it

0.510

0.138

0.703

0.178

0.264

1.021

0.126

0.170

0.236

0.130

0.220

Corr(ε it,ε it−

1)

0.951

0.663

0.923

0.806

0.762

0.924

0.602

0.733

0.859

0.658

0.693

†Thecolumnsunder

“Ave.”

reporttheaverageacross

types

usingtheestimated

mixingproportion

sπjas

weigh

ts.

19

Table 5: Bias in the measurement of productivity: Machine Industry in Japan, 1980-2008

Estimation by ClassificationJ = 5

Type 1 Type 2 Type 3 Type 4 Type 5Mean of |Biasit|Mean of |∆ωit| 0.403 0.453 0.170 0.123 0.257

Table 6: The Effect of ωit on Investment

by Classification

J = 1 J = 3 J = 5Type 1 Type 2 Type 3 Type 1 Type 2 Type 3 Type 4 Type 5

αω 0.016 0.090 -0.007 0.068 -0.016 0.094 0.054 0.010 0.083( 0.004 ) ( 0.018 ) ( 0.006 ) ( 0.014 ) ( 0.007 ) ( 0.022 ) ( 0.014 ) ( 0.013 ) ( 0.020 )

βjm 0.340 0.623 0.184 0.438 0.112 0.642 0.387 0.267 0.509

βjk/βj` 0.617 0.753 0.159 0.470 0.102 0.772 0.155 0.376 0.455

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

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and Implications,” Journal of Political Economy, 105(2): 249-283.

20

[8] Doraszelski, Ulrich, and Jordi Jaumandreu (2014) “Measuring the Bias of Technological

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A Appendix

A.1 Proof of Proposition 1

We drop the superscript j in this proof because J = 1. Let (s`t, smt ) = (lnS`t , lnS

mt ) and let

∆st := s`t − smt . From the definition of S`t and Smt , let lnWt(st, Xt) := ∆st + ln(Mt/Lt) + lnPM,t

and lnYt(st, Xt) := lnMt − smt + ln(PM,t/PY,t). Denote the density functions of (smt ,∆st, Xt) by

pt(smt ,∆st, Xt), which can be identified from Pt(St, Xt). Because E[smt |Xt] = ln (GM,t(Xt)Et[e

ε])

and E[∆st|Xt] = ln(GL,t(Xt)Et[e

ζ ])− ln (GM,t(Xt)Et[e

ε]), we have smt = E[smt |Xt]− εt and ∆st =

E[∆st|Xt] − ζt. Then, we may identify gεζ,t(·) as gεζ,t(ε, ζ) =∫pt(E[smt |Xt = X] − ε, E[∆st|Xt =

X] − ζt, X)dX. Similarly, because vt = lnWt(st, Xt) − E[lnWt(st, Xt)] − ζt = lnWt(st, Xt) −E[lnWt(st, Xt)]−(E[∆st|Xt]−∆st), we may identify gv(v) from the density function of (st, Xt). Fur-

thermore, from E[smt |Xt] = lnGM,t(Xt) + ln∫eεgε,t(ε)dε, we may identify GM,t(Xt) as GM,t(Xt) =

exp(E[smt |Xt]− ln

∫eεgε,t(ε)dε

). Similarly, GL,t(Xt) = exp

(E[∆st|Xt]− ln

∫eεgε,t(ε)dε+ ln

∫eζgζ,t(ζ)dζ

).

This proves part (a).

We proceed to prove part (b). Fix (L0,M0) ∈ L×M such that L0 < Lt and M0 < Mt. BecauseGL,t(Xt)

Lt= ∂ lnFt(Xt)

∂Ltand

GM,t(Xt)Mt

= ∂ lnFt(Xt)∂Mt

, we have

lnFt(Kt, Lt,Mt) =

∫ Lt

L0

GL,t(Kt, L,Mt)

LdL+

∫ Mt

M0

GM,t(Kt, L0,M)

MdM + lnFt(Kt, L0,M0). (23)

It follows from (1), (23), εt = E[smt |Xt]− sm, and lnYt + smt = lnMt + ln(PM,t/PY,t). that

ωt = yt(Xt; θ1)− lnFt(Kt, L0,M0), where (24)

yt(Xt; θ1) := lnMt + ln(PM,t/PY,t)−{∫ Lt

L0

GL,t(Kt, L,Mt)

LdL+

∫ Mt

M0

GM,t(Kt, L0,M)

MdM − E[smt |Xt]

}.

Substituting the right-hand side of (24) to ωt = h(ωt−1) + ηt and rearranging terms give

yt(Xt; θ1) = lnFt(Kt, L0,M0) + h (yt−1(Xt−1; θ1)− lnFt−1(Kt−1, L0,M0)) + ηt, (25)

where the second term on the right hand side only depends on Xt−1. Fix K0 ∈ K and let Ct :=

lnFt(K0, L0,M0). Then, from (25) and E[ηt|It−1] = 0, lnFt(Kt, L0,M0) is identified up to constant

Ct as

lnFt(Kt, L0,M0) = Ct + E[yt(Xt; θ1)|Kt, Xt−1]− E[yt(Xt; θ1)|Kt = K0, Xt−1]. (26)

22

It follows from the moment restriction E[ωt] = 0 with (24) and (26) that we may identify Ct as

Ct = E {yt(Xt; θ1)− E[yt(Xt; θ1)|Kt, Xt−1] + E[yt(Xt; θ1)|Kt = K0, Xt−1]} .

Therefore, lnFt(Kt, L0,M0) is identified from (26), and the identification of lnFt(Lt,Kt,Mt) for

t ≥ 2 follows from (23) given that the first two terms on the right hand side of (23) is identified

from and GL,t(Xt) and GM,t(Xt).

Finally, we prove the identification of gη(·) and h(·). Because ωt = yt(Xt; θ1)−lnFt(Kt, L0,M0),

we may identify the joint density function of ωt and ωt−1, denoted by pω(ωt, ωt−1), from the joint

distribution of (Xt, Xt−1) for t ≥ 3. Then, h(ωt−1) is identified as h(ωt−1) = Et[ωt|ωt−1] =∫ωtpω(ωt|ωt−1)dωt, where pω(ωt|ωt−1) = pω(ωt, ωt−1)/

∫pω(ωt, ωt−1)dωt, while the density function

of ηt is identified as gη(η) =∫pω(h(ωt−1) + η, ωt−1)dωt−1. This proves part (b).


The distribution of {St, Xt}Tt=1 for type j is given by

Pjt ({St, Xt}Tt=1) = Pj1(S1, X1)

T∏t=2

Pjt (St, Xt|{St−s, Xt−s}t−1s=1)

= Pj1(S1|X1)Pj1(X1)

T∏t=2

Pjt (St|Xt, {St−s, Xt−s}t−1s=1)Pjt (Xt|{St−s, Xt−s}t−1

s=1).

(27)

In view of the second and the third equations of (4), we have

Pjt (St|Xt, {St−s, Xt−s}t−1s=1) = Pjt (St|Xt). (28)

Furthermore,

Pjt (Xt|{St−s, Xt−s}t−1s=1) = Pjt (Kt, ωt, vt|{St−s,Kt−s, ωt−s, vt−s}t−1

s=1)

= Pjt (ωt, vt|Kt, {St−s,Kt−s, ωt−s, vt−s}t−1s=1)Pjt (Kt|{St−s,Kt−s, ωt−s, vt−s}t−1

s=1)

= Pjω(ωt|ωt−1)Pjv(vt)Pjt (Kt|Kt−1, ωt−1)

= Pjt (Kt, ωt, vt|Kt−1, ωt−1)

= Pjt (Kt, ωt, vt|Kt−1, ωt−1, vt−1)

= Pjt (Xt|Xt−1),

(29)

where the first equality and the last equality hold because there is a one-to-one mapping between

Xt and (Kt, ωt, vt) in view of Assumption 4(b), the third equality follows from Assumptions 2 and

3, the second to the last equality holds because vt is i.i.d.. Therefore, the stated result follows from

(27)-(29).

23


We apply the argument of Kasahara and Shimotsu (2009) and Hu and Shum (2012) under the

assumption that unobserved heterogeneity is permanent and discrete. The proof is constructive.

Consider the case that T = 4. Fix (Z2, Z3) at (Z2, Z3) and choose (Z2, Z3) ∈ Z2 × Z3,

(a1, ..., aJ) ∈ ZJ1 and (b1, ..., bJ−1) ∈ ZJ−14 that satisfy Assumption 7. Evaluating (7) at (Z2, Z3) =

(Z2, Z3) gives

P ({Zt}4t=1) =J∑j=1

πjP j4 (Z4|Z3)P j3 (Z3|Z2)P j2 (Z2|Z1)P j1 (Z1)

=

J∑j=1

λj4(Z4|Z3)λj3(Z3|Z2)λj2(Z1, Z2),

(30)

where λj4(Z4|Z3) := P j4 (Z4|Z3 = Z3), λj3(Z3|Z2) := P j3 (Z3 = Z3|Z2 = Z2), and λj2(Z1, Z2) :=

πjP j2 (Z2 = Z2|Z1)P j1 (Z1). Integrating out Z4 from (30) gives

P ({Zt}3t=1) =J∑j=1

λj3(Z3|Z2)λj2(Z1, Z2). (31)

Let fZ2,Z3(a, b) := P ((Z1, Z2, Z3, Z4) = (a, Z2, Z3, b)) and fZ2,Z3(a) := P ((Z1, Z2, Z3) = (a, Z2, Z3)).

Evaluating (30) at Z1 = a1, ..., aJ and Z4 = b1, ..., bJ−1 gives M(M − 1) equations while evaluating

(31) at Z1 = a1, ..., aJ gives M equations. Collecting these M(M − 1) + M = M2 equations and

denoting them using matrix notation, we have

PZ2,Z3 = L′z3DZ3|Z2Lz2 , (32)

where

PZ2,Z3 :=

fZ2,Z3(a1) fZ2,Z3(a2) · · · fZ2,Z3(aJ)

fZ2,Z3(a1, b1) fZ2,Z3(a2, b1) · · · fZ2,Z3(aJ , b1)...

... . . ....

fZ2,Z3(a1, bJ−1) fZ2,Z3(a2, b1) · · · fZ2,Z3(aJ , bJ−1)

,

Lz3 :=

1 λ1

4(b1|Z3) · · · λ14(bJ−1|Z3)

...... . . .

...

1 λJ4 (b1|Z3) · · · λJ4 (bJ−1|Z3)

, Lz2 :=

λ1

2(a1, Z2) · · · λ12(aJ , Z2)

...... . . .

λJ2 (a1, Z2) · · · λJ2 (aJ , Z2)

,(33)

and DZ3|Z2:= diag

(λ1

3(Z3|Z2), ..., λJ3 (Z3|Z2)). Evaluating (32) at four different points, (Z2, Z3),

(Z2, Z3), (Z2, Z3), and (Z2, Z3) gives

PZ2,Z3 = L′z3DZ3|Z2Lz2 , PZ2,Z3

= L′z3DZ3|Z2Lz2 ,

PZ2,Z3= L′z3DZ3|Z2

Lz2 , PZ2,Z3= L′z3DZ3|Z2

Lz2 .

24

Then, under Assumption 7,

A := PZ2,Z3(PZ2,Z3)−1PZ2,Z3

(PZ2,Z3)−1 = L′z3DZ2,Z2,Z3,Z3

(L′z3)−1,

where

DZ2,Z2,Z3,Z3:= DZ3|Z2

(DZ3|Z2)−1DZ3|Z2

(DZ3|Z2)−1. (34)

Because AL′z3 = L′z3DZ2,Z2,Z3,Z3, the eignvalues of A determine the diagonal elements of

DZ2,Z2,Z3,Z3while the right eigenvectors of A determine the columns of L′z3 up to multiplicative

constant. Denote the right eigenvectors of A by L′z3C, where C is some diagonal matrix. Now we

can determine the diagonal matrix DZ2,Z2,Z3,Z3C from the first row of AL′z3C = L′z3DZ2,Z2,Z3,Z3

C

because the first row of L′z3 is a vector of ones. Then, L′z3 is determined uniquely from AL′z3C and

DZ2,Z2,Z3,Z3C as L′z3 = (AL′z3C)(DZ2,Z2,Z3,Z3

C)−1 in view of AL′z3 = L′z3DZ2,Z2,Z3,Z3. Therefore,

Lz3 is identified. Repeating the above argument for all values of Z3 ∈ Z3 identifies {P j4 (Z4|Z3 =

Z3)}Jj=1 for each Z3 ∈ Z3 for Z4 = (b1, ..., bJ−1) that satisfies Assumption 7(a).

Evaluating P (Z4, Z3|Z2) at (Z2, Z3) = (Z2, Z3), we have

P (Z4, Z3 = Z3|Z2 = Z2) =

J∑j=1

πjZ2P j4 (Z4|Z3)P j3 (Z3|Z2) =

J∑j=1

λj4(Z4|Z3)λj3(Z3|Z2), (35)

where πjZ2:=

πjP j2 (Z2=Z2)P2(Z2=Z2) and λj3(Z3|Z2) := πjZ2

P j3 (Z3 = Z3|Z2 = Z2). Then, evaluating (35)

at Z4 = b1, ..., bJ−1 and collecting them into a vector together with P (Z3 = Z3|Z2 = Z2) =∑Jj=1 λ

j3(Z3|Z2) gives

pZ3|Z2= L′z3dZ3|Z2

,

where dZ3|Z2= (λ1

3(Z3|Z2), ...., λJ3 (Z3|Z2))′ and pZ3|Z2= (P (Z3 = Z3|Z2 = Z2), P ((Z4, Z3) =

(b1, Z3)|Z2 = Z2), ..., P ((Z4, Z3) = (bJ−1, Z3)|Z2 = Z2))′. Therefore, we uniquely determine

πjZ2P j3 (Z3 = Z3|Z2 = Z2) from dZ3|Z2

= (L′z3)−1pZ3|Z2. Repeating the above argument across

all possible values of (Z2, Z3) ∈ Z2 × Z3 determines the value of πjZ2P j3 (Z3 = Z3|Z2 = Z2)

for every (Z2, Z3) ∈ Z2 × Z3. Then, πjZ2and P j3 (Z3 = Z3|Z2 = Z2) are uniquely identified as

πjZ2=∫Z3πjZ2

P j3 (Z3|Z2 = Z2)dZ3 and P j(Z3 = Z3|Z2 = Z2) = [πjZ2P j3 (Z3 = Z3|Z2 = Z2)]/πjZ2

.

Therefore, {P j3 (Z3|Z2)}Jj=1 is identified.

Evaluating P j3 (Z3|Z2) at (Z2, Z3) = (Z3, Z2) for j = 1, ..., J identifies DZ3|Z2and, from (32),

Lz2 is identified as Lz2 = (DZ3|Z2)−1(L′z3)−1PZ2,Z3 . Once DZ3|Z2

and Lz2 are identified, we

can determine Lz3(ζ) = (λ14(ζ|Z3), ..., λJ4 (ζ|Z3))′ for any ζ ∈ Z4 by constructing pZ2,Z3(ζ) =

(fZ2,Z3(a1, ζ), fZ2,Z3(a2, ζ), ..., fZ2,Z3(aJ , ζ)) from the observed data, and using the relationship

Lz3(ζ) = (DZ3|Z2)−1(L′z2)−1pZ2,Z3(ζ)′. Similarly, we can determine Lz2(ξ) = (λ1

2(ξ, Z2), ..., λJ2 (ξ, Z2))′

for any ξ ∈ Z1 by constructing pZ2,Z3(ξ) = (fZ2,Z3(ξ), fZ2,Z3(ξ, b1), fZ2,Z3(ξ, b2), ..., fZ2,Z3(ξ, bJ−1))′

and using the relationship Lz2(ξ) = (DZ3|Z2)−1(L′z3)−1pZ2,Z3(ξ). Therefore, {P j4 (Z4|Z3 = Z3), πjP j2 (Z2 =

Z2|Z1)P j1 (Z1)}Jj=1 is identified. Repeating this argument for all possible values of (Z2, Z3) ∈

25

Z2 × Z3 identifies {P j4 (Z4|Z3), πjP j2 (Z2|Z1)P j1 (Z1)}Jj=1. Finally, {πj , P j2 (Z2|Z1), P j1 (Z1)}Jj=1 is

identified from {πjP j2 (Z2|Z1)P j1 (Z1)}Jj=1 as πj =∫Z1

∫Z2

[πjP j2 (Z2|Z1)P j1 (Z1)]dZ2dZ1, P j1 (Z1) =

[∫Z2

[πjP j2 (Z2|Z1)P j1 (Z1)]dZ2]/πj , and P j2 (Z2|Z1) = [πjP j2 (Z2|Z1)P j1 (Z1)]/[πj×P j1 (Z1)]. This proves

the stated result.


We first show that PH,t and {ψjt }Jj=1 are identified from {πj , P jt (Wt)}Jj=1. Because Ejt [lnWt] =

ln(PH,teψjt ), we may have ψjt = Ejt [lnWt]− PH,t for j = 1, ..., J , where Ejt [lnWt] is identified from

P jt (Wt). Then, PH,t is identified from∑J

j=1 πjeE

jt [lnWt]−lnPH,t = 1 as lnPH,t = ln

(∑Jj=1 π

jeEjt [lnWt]

).

Once PH,t and {ψjt }Jj=1 are identified, then repeating the argument in the proof of Proposition 1

for each type proves the stated result.


Consider i ∈ Ij so that j = j∗(i). For each T , let πjT := πj∗L1i(αj∗m ,σ

j∗ε ;T )∑J

k=1 πk∗L1i(αk∗m ,σk∗ε ;T )

, where (πj∗, αj∗m , σj∗ε )

is the true value of (πj , αjm, σjε ). Then,

πji − 1 = (πji − πjT ) + (πjT − 1). (36)

For the first term, πji − πjT = Op(N

−1/2) as N → ∞ because the maximum likelihood estimator

(πj , αjm, σjε ) is a root-N consistent estimator of (πj∗, αj∗m , σ

j∗ε ) when the number of components J is

correctly specified.

For the second term of (36), define ξjkit := lnL1it(αj∗m , σ

j∗ε )−lnL1it(α

k∗m , σ

k∗ε ) and ajk := E[ξjkit |i ∈

Ij ] > 0, and we have

πjT =1

1 +∑

k 6=j (πk∗/πj∗) exp(−∑T

t=1 ξjkit

) . (37)

For i ∈ Ij , k 6= j,

exp

(−

T∑t=1

ξjkit

)=

{exp

(−

T∑t=1

ξjkit

)− exp(−ajkT )

}+ exp(−ajkT )

= exp(−ajkT )

{exp

(−

T∑t=1

(ξjkit − ajk)

)− 1

}︸︷︷︸

Op(T 1/2)

+ exp(−ajkT )

= Op

(exp(−ajkT )T 1/2

)as T → ∞. It follows that

∑k 6=j

(πk

∗/πj∗

)exp

(−∑T

t=1 ξjkit

)is Op

(exp(−ajT )T 1/2

), where aj :=

mink 6=j ajk. Therefore, in view of (37), the consistency of πjT as T →∞ and the mean value theorem

26

give

πjT − 1 = Op

(exp(−ajT )T 1/2

). (38)

Then, the stated result follows from (36), (38), and πji−πjT = op(N

−1/2) becauseOp(exp(−ajT )T 1/2

)=

op(N−1/2) as N,T →∞ under Assumption 10.

A.6 Assumption 7 under Cobb-Douglas production function

In the following, we discuss the conditions under which Assumption 7 holds when the production

function is Cobb-Douglas.

Example 1 (continued). For random coefficients model (8), we may write Lz3, Lz2, and DZ3|Z2as

follows. Throughout the analysis, we fix the value of {Yt}Tt=1 at, say, {yt}Tt=1 so that the variation in

the values of aj’s and bj’s are due the variation in the values of Z1 and Z4. Denote Z3 = (y3, s3, x3)

and bh = (y3, bsh, x4) for h = 1, ..., J − 1. Then,

λjZ3

(bh) = P j4 (S4 = bsh|X4 = x4)P j4 (X4 = x4|X3 = x3) = cj4gjε (ln(αjm,4E

j)− ln bsh),

where cj4 = P j4 (X4 = x4|X3 = x3). Therefore, we have

Lz3 = diag{c14, ...., c

J4 }

1 g1

ε (ln(α1m,4E1)− ln bs1) · · · g1

ε (ln(α1m,4E1)− ln bsJ−1)

...... . . .

...

1 gJε (ln(αJm,4EJ)− ln bs1) · · · gJε (ln(αJm,4EJ)− ln bsJ−1)

.Similarly, denote Z2 = (s2, x2) and ah = (ash, x1) for h = 1, ..., J . Then,

λjZ2

(ah) = P j2 (S2 = s2|X2 = x2)P j1 (S1 = ash|X1 = x1)P j1 (X1 = x1) = cj2gjε (ln a

sh − ln(αjmE)),

where cj2 = P j2 (S2 = s2|X2 = x2)P j1 (X1 = x1). Then, we have

Lz2 = diag{c12, ...., c

J2 }

g1ε (ln(α1

m,1E1)− ln as1) · · · g1ε (ln(α1

m,1E1)− ln asJ)... . . .

...

gJε (ln(αJm,1EJ)− ln as1) · · · gJε (ln(αJm,1EJ)− ln asJ)

.For Assumption 7(a), we choose x4, x3, x2, x1, and s2 so that cj2 6= 0 and cj3 6= 0 for any j

and find (as1, ..., asJ) and (bs1, ..., b

sJ−1) such that Lz3 and Lz2 are nonsingular. Because each point

of (as1, ..., asJ) and (bs1, ..., b

sJ−1) refers to a value of lnS1 and lnS4, the full rank condition of Lz3

and Lz2 holds if the value of probability density function of lnS1 and lnS4 changes heterogeneously

across types when we change the value of lnS1 and lnS4.

27

Let Z3 = (s3, x3) and Z2 = (s2, x2). Then,

λj(Z3|Z2) = πjgjε (ln s3 − ln(αjm,3Ej))P j3 (X3 = x3|X2 = x2). (39)

Pick Z3 = (s3, x3) and Z2 = (s2, x2). Assumption 7(b) holds if P j3 (x3|x2) 6= 0 and P j3 (x3|x2) 6= 0

for all j. Then, we have

DZ3|Z2(DZ3|Z2

)−1DZ3|Z2(DZ3|Z2

)−1 = diag

{P 1

3 (x3|x2)

P 13 (x3|x2)

P 13 (x3|x2)

P 13 (x3|x2)

, ...,P J3 (x3|x2)

P J3 (x3|x2)

P J3 (x3|x2)

P J3 (x3|x2)

}.

Therefore, Assumption 7(c) requires thatP j3 (x3|x2)

P j3 (x3|x2)

P j3 (x3|x2)

P j3 (x3|x2)takes different values across different

j’s, namely, the transition probability of X3 given X2 changes heterogeneously across types when

we change the value of X2.

28

Figure 1: Histogram ofPM,tMit

PY,tYit

0D

ensity

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Intermediate Share

Figure 2: Histogram of(PM,tMit

PY,tYit

)i

over 28 years

0D

ensity

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Average Intermediate Share

Figure 3:PM,tMit

PM,tMit+WtLit

0D

en

sity

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1PM/(PM+WL)

Figure 4:(

PM,tMit

PM,tMit+WtLit

)i

over 28 yrs

0D

en

sity

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Average PM/(PM+WL)

29

Figure 5: Trends in the log ofPM,tMit

PY,tYit, output, and inputs in Machine industry

lnPM,tMit

PY,tYitlnYit

-5.0

-2.5

0.0

1980 1990 2000year

lnmY_it

12.5

15.0

17.5

20.0

1980 1990 2000year

y_it

lnKit lnLit

12

14

16

18

20

1980 1990 2000year

k_it

3

6

9

1980 1990 2000year

l_it

lnMit

9

12

15

18

21

1980 1990 2000year

m_it

Notes: This figure shows each firms’ inputs and outputs in each year. Each line represents a different firm.

30

Figure 6: Trends inPM,tMit

PY,tYitfor subindustries in Machine industry

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

2511: Boiler prime mover 2521: Metal machine tools 2522: Metal working machinery

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

2523: Machinery tool 2531: Textile machinery 2532: Agricultural machines

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

2533: Construction and mining equipment 2534: Chemical machinery 2535: Office machinery

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

2536: Special industrial machinery 2537: General industrial machinery 2541: General Mechanical Components

Figure 7: Posterior probabilities for J = 3 and J = 5

J = 3posterior.1 posterior.2 posterior.3

0

50

100

150

200

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8value

count

J = 5posterior.1 posterior.2 posterior.3 posterior.4 posterior.5

0

50

100

150

200

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8value

count

31


PY,tYitand lnYit in Machine industry for J = 3

PM,tMit

PY,tYit

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

Type 1 2 3

lnYit

12.5

15.0

17.5

20.0

1980 1990 2000year

y_it

Type 1 2 3

32


PY,tYitin subindustries when J = 3

2511 2521 2522 2523

2529 2531 2532 2533

2534 2535 2536 2537

2541

0.00

0.25

0.50

0.75

1.00

0.00

0.25

0.50

0.75

1.00

0.00

0.25

0.50

0.75

1.00

0.00

0.25

0.50

0.75

1.00

1980 1990 2000year

exp(lnmY_it)

Type 1 2 3

33

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